Advertisements
Advertisements
प्रश्न
If sin θ + cos θ = x, prove that `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`
Advertisements
उत्तर
Given `sin theta + cos theta = x`
Squaring the given equation, we have
`(sin theta + cos theta)^2 = x^2`
`=> sin^2 theta + 2 sin theta cos theta = cos^2 theta = x^2`
`=> (sin^2 theta + cos^2 theta) + 2sin theta cos theta = x^2`
`=> 1 + 2 sin theta cos theta = x^2`
`=> 2 sin theta cos theta = x^2 -1`
`=> sin theta cos theta = (x^2- 1)/2`
Squaring the last equation, we have
`(sin theta cos theta)^2 = (x^2 - 1)^2/4`
`=> sin^2 theta cos^2 theta = (x^2 - 1)^2/4`
`=> sin^2 theta cos^2 theta = (s^2 -1)/4`
Therefore, we have
`sin^6 theta + cos^6 theta = (sin^2 theta)^3 + (cos^2 theta)^3`
`= (sin^2 theta + cos^2 theta)^3 - 3sin^3 theta cos^2 theta (sin^2 theta + cos^2 theta)`
`= (1)^3 - 3 ((x^2 - 1)^2)/4 (1)`
`= 1 - 3 (x^2 - 1)^2/4 (1)`
`x = 1 - 3 (x^2 - 1)^2/4`
`= (4- 3(x^2 - 1)^2)/4`
hence Proved
APPEARS IN
संबंधित प्रश्न
If `x/a=y/b = z/c` show that `x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)`.
Prove the following trigonometric identities.
`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`
Prove the following trigonometric identities.
`tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ`
Prove the following trigonometric identities.
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
Prove the following identities:
cot2 A – cos2 A = cos2 A . cot2 A
Prove the following identities:
(cosec A + sin A) (cosec A – sin A) = cot2 A + cos2 A
If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that : x2 + y2 + z2 = r2
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
`sqrt((1 + sin θ)/(1 - sin θ)) = sec θ + tan θ`
Write the value of `(sin^2 theta 1/(1+tan^2 theta))`.
Write the value of ` sec^2 theta ( 1+ sintheta )(1- sintheta).`
Write the value of tan10° tan 20° tan 70° tan 80° .
9 sec2 A − 9 tan2 A is equal to
Prove the following identity :
`cosec^4A - cosec^2A = cot^4A + cot^2A`
Prove the following identity :
`(cos^3θ + sin^3θ)/(cosθ + sinθ) + (cos^3θ - sin^3θ)/(cosθ - sinθ) = 2`
Evaluate:
`(tan 65°)/(cot 25°)`
Prove that cos θ sin (90° - θ) + sin θ cos (90° - θ) = 1.
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
If 5 tan β = 4, then `(5 sin β - 2 cos β)/(5 sin β + 2 cos β)` = ______.
`1/sin^2θ - 1/cos^2θ - 1/tan^2θ - 1/cot^2θ - 1/sec^2θ - 1/("cosec"^2θ) = -3`, then find the value of θ.
